Boyer and Moore have discussed a recursive function that puts conditional
expressions into normal form [1]. It is difficult to prove that this function
terminates on all inputs. Three termination proofs are compared: (1) using a
measure function, (2) in domain theory using LCF, (3) showing that its
recursion relation, defined by the pattern of recursive calls, is well-founded.
The last two proofs are essentially the same though conducted in markedly
different logical frameworks. An obviously total variant of the normalize
function is presented as the `computational meaning' of those two proofs. A
related function makes nested recursive calls. The three termination proofs
become more complex: termination and correctness must be proved simultaneously.
The recursion relation approach seems flexible enough to handle subtle
termination proofs where previously domain theory seemed essential
